General Theory of Statistical Fluctuations with Applications to Metastable states, Nernst Points, and Compressible Multi-component Mixtures

Abstract

The general fluctuation theory is reviewed with special attention to the role played by different ensembles, and is extended to incorporate stationary metastable states obtained in the long time limit. The fluctuation in a quantity depends on the nature of the ensemble and contains at most n different fluctuation contributions, where is the number of fluctuating extensive quantities in the ensemble. We prove four general theorems and a corollary for statistical fluctuations valid for any thermodynamic system. We also demonstrate by two examples that the results of the theory remain valid regardless of the magnitude of the fluctuations. To avoid certain physical paradoxes, it is postulated that stationary metastable states like the ideal glass cannot exist in Nature. We also prove a generalized Nernst theorem valid at Nernst points at which certain susceptibility like the heat capacity vanishes. The theorem is no longer restricted to absolute temperature. We calculate statistical fluctuations in the number of monomers and other physical quantities of interest in a compressible mixture. We demonstrate that the density and composition fluctuations are in general not statistically independent, which is contrary to some recent claims. The standard isothermal compressibility at constant monomer numbers does not represent the density fluctuation in all ensembles. We show that the density fluctuation at constant composition is a meaningless concept, except at absolute zero. We prove a relation between the weighted monomer number fluctuation and the volume fluctuation in a multi-component system, which is an extension of a well-known similar relation for a single component system.